Since \(p(Ax)=p(x)\)  for all \(x \in \mathbb {R}^{n}\) , a change of variable gives
\(\mathcal {A}\left( \left\lbrace  \lambda _{j}\right\rbrace  _{j=1}^{\infty },\left\lbrace Q_{j}^{\ast }\right\rbrace  _{j=1}^{\infty },p(.)\right) =\left\Vert \left\lbrace \sum \limits _{j=1}^{\infty }\left( \frac{\lambda _{j}\chi _{Q_{j}^{\ast }}}{\left\Vert \chi _{Q_{j}\ast }\right\Vert _{p(.)}}\right) ^{\underline{p}}\right\rbrace  ^{\frac{1}{\underline{p}}}\right\Vert _{p(.)}\) 
\(= \left\Vert \left\lbrace \sum \limits _{j=1}^{\infty }\left( \frac{\lambda _{j}\chi _{A^{-1}Q_{j}^{\ast }}}{\left\Vert \chi _{A^{-1}Q_{j}^{\ast }}\right\Vert _{p(.)}}\right) ^{\underline{p}}\right\rbrace  ^{\frac{1}{\underline{p}}}\right\Vert _{p(.)} = : \mathcal {A}\left( \left\lbrace  \lambda _{j}\right\rbrace  _{j=1}^{\infty },\left\lbrace A^{-1}Q_{j}^{\ast }\right\rbrace  _{j=1}^{\infty },p(.)\right),\) 
it is easy to check that \(Q_j \subset A^{-1}Q_{j}^{\ast }\)  for all \(j\) . Moreover, there exists a positive universal constant \(c\)  such that
\(|A^{-1}Q_{j}^{\ast }| \le c |Q_j|\)  for all \(j\) . The same argument utilized in the proof of Lemma 4.8 in [1]} works in this case, so
\(\mathcal {A}\left( \left\lbrace  \lambda _{j}\right\rbrace  _{j=1}^{\infty },\left\lbrace A^{-1}Q_{j}^{\ast }\right\rbrace  _{j=1}^{\infty },p(.)\right) \lesssim \mathcal {A}\left( \left\lbrace  \lambda _{j}\right\rbrace  _{j=1}^{\infty },\left\lbrace Q_{j}\right\rbrace  _{j=1}^{\infty }, p(.)\right).\) 
The proof is therefore concluded.
